The basic arithmetic operations (i.e., addition, subtraction, and multiplication) in the prime finite field GF(p) have numerous applications in cryptography, such as decipherment in RSA systems, Diffie-Hellman key exchange, elliptic curve cryptography, the Digital Signature Standard (DSS), and the elliptic curve digital signature algorithm (ECDSA). These applications demand high-speed software and hardware implementations of the arithmetic operations in the field GF(p), typically for p in a range such that 160≦┌log2(p)┐≦2048. Improved methods and apparatus are needed for these and other applications.